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Consider (for all $t\geq 0$) a linear time transformation function $\nu(t)=at+b$ with the following properties:

  1. $\nu(0)=-1$
  2. $\nu(t)$ is an increasing function of the time index $t$ i.e. $a>0$.
  3. $\nu(t)$ is a continuous function of the time index $t$.
  4. $\nu(t)$ is a 1-1 function of time index $t$.
  5. $\nu(t+1)$-$\nu(t)=a$ for all $t\geq 0$.

Q: Can we define a Lévy process with $\nu(t)$ as the time index (rather than $t$) or do we need more conditions on $\nu(t)$? In other words, I would like to see the possibility of defining $\{Z_{\nu(t)};t\geq 0\}$, where $Z_{\nu(t)}$ is a Lévy process.

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  • $\begingroup$ It is a little unclear what you are trying to do. First, what do you mean that the index $t$ "is a continuous function"? What is it a function of? Second, although transforming the index creates a new process, it usually is not a Lévy process, because it differentially changes the durations. Are you asking what functions $T$ will transform one Lévy process into another? $\endgroup$
    – whuber ♦
    Mar 4, 2015 at 22:43
  • $\begingroup$ I editted the question to make it more clear. $\endgroup$
    – Stat
    Mar 5, 2015 at 13:22
  • $\begingroup$ The original $Z_{t}$ was a Levy process itself? $\endgroup$
    – Cristián Antuña
    Mar 5, 2015 at 16:04
  • $\begingroup$ @ Cristián Antuña : no, $Z_t$ is not necessarily a Levy process. If this is the case, then I know that $Z_{\nu(t)}$ will be an additive process and not necessarily a Levy process as said in Example 14.5 here. Normally, when we want to define a Levy process, we consider a time index $t\geq 0$. I just wanted to know the possibility of defining $Z_{\nu(t)}$. $\endgroup$
    – Stat
    Mar 5, 2015 at 16:14
  • $\begingroup$ What are the requirements on $Z_t$? $\endgroup$
    – KOE
    Mar 29, 2015 at 19:00

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